Se da la desigualdad:
$$\sqrt{- x^{2} + 3 x} \cos{\left(x - \sin{\left(x \right)} \right)} \geq 0$$
Para resolver esta desigualdad primero hay que resolver la ecuación correspondiente:
$$\sqrt{- x^{2} + 3 x} \cos{\left(x - \sin{\left(x \right)} \right)} = 0$$
Resolvemos:
$$x_{1} = -2.30988146001006$$
$$x_{2} = -79.3715275333246$$
$$x_{3} = 35.3892303830675$$
$$x_{4} = -21.1594373815488$$
$$x_{5} = 71.4249198389855$$
$$x_{6} = -91.9378981476837$$
$$x_{7} = 0.916089372113664 + 1.8093613412957 i$$
$$x_{8} = 14.8762520743692$$
$$x_{9} = 2.30988146001006$$
$$x_{10} = -16.5396744615287$$
$$x_{11} = -85.6547128405042$$
$$x_{12} = -54.2387863046062$$
$$x_{13} = 96.5576610677038$$
$$x_{14} = 41.672415690247$$
$$x_{15} = 33.725807995908$$
$$x_{16} = -41.672415690247$$
$$x_{17} = -98.2210834548633$$
$$x_{18} = -83.9912904533447$$
$$x_{19} = -46.2921786102672$$
$$x_{20} = 98.2210834548633$$
$$x_{21} = -3.97330384716953$$
$$x_{22} = -66.8051569189654$$
$$x_{23} = 40.0089933030876$$
$$x_{24} = 3.97330384716953$$
$$x_{25} = 54.2387863046062$$
$$x_{26} = -10.2564891543491$$
$$x_{27} = -40.0089933030876$$
$$x_{28} = -96.5576610677038$$
$$x_{29} = 27.4426226887284$$
$$x_{30} = 47.9556009974266$$
$$x_{31} = -29.1060450758879$$
$$x_{32} = 10.2564891543491$$
$$x_{33} = 83.9912904533447$$
$$x_{34} = 91.9378981476837$$
$$x_{35} = -27.4426226887284$$
$$x_{36} = 0.916089372113664 - 1.8093613412957 i$$
$$x_{37} = 52.5753639174468$$
$$x_{38} = 79.3715275333246$$
$$x_{39} = 66.8051569189654$$
$$x_{40} = 85.6547128405042$$
$$x_{41} = 0$$
$$x_{42} = -8.59306676718964$$
$$x_{43} = 16.5396744615287$$
$$x_{44} = -47.9556009974266$$
$$x_{45} = 22.8228597687083$$
$$x_{46} = -0.916089372113664 - 1.8093613412957 i$$
$$x_{47} = -33.725807995908$$
$$x_{48} = -90.2744757605243$$
$$x_{49} = 90.2744757605243$$
$$x_{50} = -52.5753639174468$$
$$x_{51} = -60.5219716117858$$
$$x_{52} = -77.7081051461651$$
$$x_{53} = -73.088342226145$$
$$x_{54} = -35.3892303830675$$
$$x_{55} = 46.2921786102672$$
$$x_{56} = 8.59306676718964$$
$$x_{57} = 77.7081051461651$$
$$x_{58} = -71.4249198389855$$
$$x_{59} = 58.8585492246263$$
$$x_{60} = 60.5219716117858$$
Descartamos las soluciones complejas:
$$x_{1} = -2.30988146001006$$
$$x_{2} = -79.3715275333246$$
$$x_{3} = 35.3892303830675$$
$$x_{4} = -21.1594373815488$$
$$x_{5} = 71.4249198389855$$
$$x_{6} = -91.9378981476837$$
$$x_{7} = 14.8762520743692$$
$$x_{8} = 2.30988146001006$$
$$x_{9} = -16.5396744615287$$
$$x_{10} = -85.6547128405042$$
$$x_{11} = -54.2387863046062$$
$$x_{12} = 96.5576610677038$$
$$x_{13} = 41.672415690247$$
$$x_{14} = 33.725807995908$$
$$x_{15} = -41.672415690247$$
$$x_{16} = -98.2210834548633$$
$$x_{17} = -83.9912904533447$$
$$x_{18} = -46.2921786102672$$
$$x_{19} = 98.2210834548633$$
$$x_{20} = -3.97330384716953$$
$$x_{21} = -66.8051569189654$$
$$x_{22} = 40.0089933030876$$
$$x_{23} = 3.97330384716953$$
$$x_{24} = 54.2387863046062$$
$$x_{25} = -10.2564891543491$$
$$x_{26} = -40.0089933030876$$
$$x_{27} = -96.5576610677038$$
$$x_{28} = 27.4426226887284$$
$$x_{29} = 47.9556009974266$$
$$x_{30} = -29.1060450758879$$
$$x_{31} = 10.2564891543491$$
$$x_{32} = 83.9912904533447$$
$$x_{33} = 91.9378981476837$$
$$x_{34} = -27.4426226887284$$
$$x_{35} = 52.5753639174468$$
$$x_{36} = 79.3715275333246$$
$$x_{37} = 66.8051569189654$$
$$x_{38} = 85.6547128405042$$
$$x_{39} = 0$$
$$x_{40} = -8.59306676718964$$
$$x_{41} = 16.5396744615287$$
$$x_{42} = -47.9556009974266$$
$$x_{43} = 22.8228597687083$$
$$x_{44} = -33.725807995908$$
$$x_{45} = -90.2744757605243$$
$$x_{46} = 90.2744757605243$$
$$x_{47} = -52.5753639174468$$
$$x_{48} = -60.5219716117858$$
$$x_{49} = -77.7081051461651$$
$$x_{50} = -73.088342226145$$
$$x_{51} = -35.3892303830675$$
$$x_{52} = 46.2921786102672$$
$$x_{53} = 8.59306676718964$$
$$x_{54} = 77.7081051461651$$
$$x_{55} = -71.4249198389855$$
$$x_{56} = 58.8585492246263$$
$$x_{57} = 60.5219716117858$$
Las raíces dadas
$$x_{16} = -98.2210834548633$$
$$x_{27} = -96.5576610677038$$
$$x_{6} = -91.9378981476837$$
$$x_{45} = -90.2744757605243$$
$$x_{10} = -85.6547128405042$$
$$x_{17} = -83.9912904533447$$
$$x_{2} = -79.3715275333246$$
$$x_{49} = -77.7081051461651$$
$$x_{50} = -73.088342226145$$
$$x_{55} = -71.4249198389855$$
$$x_{21} = -66.8051569189654$$
$$x_{48} = -60.5219716117858$$
$$x_{11} = -54.2387863046062$$
$$x_{47} = -52.5753639174468$$
$$x_{42} = -47.9556009974266$$
$$x_{18} = -46.2921786102672$$
$$x_{15} = -41.672415690247$$
$$x_{26} = -40.0089933030876$$
$$x_{51} = -35.3892303830675$$
$$x_{44} = -33.725807995908$$
$$x_{30} = -29.1060450758879$$
$$x_{34} = -27.4426226887284$$
$$x_{4} = -21.1594373815488$$
$$x_{9} = -16.5396744615287$$
$$x_{25} = -10.2564891543491$$
$$x_{40} = -8.59306676718964$$
$$x_{20} = -3.97330384716953$$
$$x_{1} = -2.30988146001006$$
$$x_{39} = 0$$
$$x_{8} = 2.30988146001006$$
$$x_{23} = 3.97330384716953$$
$$x_{53} = 8.59306676718964$$
$$x_{31} = 10.2564891543491$$
$$x_{7} = 14.8762520743692$$
$$x_{41} = 16.5396744615287$$
$$x_{43} = 22.8228597687083$$
$$x_{28} = 27.4426226887284$$
$$x_{14} = 33.725807995908$$
$$x_{3} = 35.3892303830675$$
$$x_{22} = 40.0089933030876$$
$$x_{13} = 41.672415690247$$
$$x_{52} = 46.2921786102672$$
$$x_{29} = 47.9556009974266$$
$$x_{35} = 52.5753639174468$$
$$x_{24} = 54.2387863046062$$
$$x_{56} = 58.8585492246263$$
$$x_{57} = 60.5219716117858$$
$$x_{37} = 66.8051569189654$$
$$x_{5} = 71.4249198389855$$
$$x_{54} = 77.7081051461651$$
$$x_{36} = 79.3715275333246$$
$$x_{32} = 83.9912904533447$$
$$x_{38} = 85.6547128405042$$
$$x_{46} = 90.2744757605243$$
$$x_{33} = 91.9378981476837$$
$$x_{12} = 96.5576610677038$$
$$x_{19} = 98.2210834548633$$
son puntos de cambio del signo de desigualdad en las soluciones.
Primero definámonos con el signo hasta el punto extremo izquierdo:
$$x_{0} \leq x_{16}$$
Consideremos, por ejemplo, el punto
$$x_{0} = x_{16} - \frac{1}{10}$$
=
$$-98.2210834548633 + - \frac{1}{10}$$
=
$$-98.3210834548633$$
lo sustituimos en la expresión
$$\sqrt{- x^{2} + 3 x} \cos{\left(x - \sin{\left(x \right)} \right)} \geq 0$$
$$\sqrt{- \left(-98.3210834548633\right)^{2} + \left(-98.3210834548633\right) 3} \cos{\left(-98.3210834548633 - \sin{\left(-98.3210834548633 \right)} \right)} \geq 0$$
16.2518727724411*I >= 0
Entonces
$$x \leq -98.2210834548633$$
no se cumple
significa que una de las soluciones de nuestra ecuación será con:
$$x \geq -98.2210834548633 \wedge x \leq -96.5576610677038$$
_____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____
/ \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
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x16 x27 x6 x45 x10 x17 x2 x49 x50 x55 x21 x48 x11 x47 x42 x18 x15 x26 x51 x44 x30 x34 x4 x9 x25 x40 x20 x1 x39 x8 x23 x53 x31 x7 x41 x43 x28 x14 x3 x22 x13 x52 x29 x35 x24 x56 x57 x37 x5 x54 x36 x32 x38 x46 x33 x12 x19
Recibiremos otras soluciones de la desigualdad pasando al polo siguiente etc.
etc.
Respuesta:
$$x \geq -98.2210834548633 \wedge x \leq -96.5576610677038$$
$$x \geq -91.9378981476837 \wedge x \leq -90.2744757605243$$
$$x \geq -85.6547128405042 \wedge x \leq -83.9912904533447$$
$$x \geq -79.3715275333246 \wedge x \leq -77.7081051461651$$
$$x \geq -73.088342226145 \wedge x \leq -71.4249198389855$$
$$x \geq -66.8051569189654 \wedge x \leq -60.5219716117858$$
$$x \geq -54.2387863046062 \wedge x \leq -52.5753639174468$$
$$x \geq -47.9556009974266 \wedge x \leq -46.2921786102672$$
$$x \geq -41.672415690247 \wedge x \leq -40.0089933030876$$
$$x \geq -35.3892303830675 \wedge x \leq -33.725807995908$$
$$x \geq -29.1060450758879 \wedge x \leq -27.4426226887284$$
$$x \geq -21.1594373815488 \wedge x \leq -16.5396744615287$$
$$x \geq -10.2564891543491 \wedge x \leq -8.59306676718964$$
$$x \geq -3.97330384716953 \wedge x \leq -2.30988146001006$$
$$x \geq 0 \wedge x \leq 2.30988146001006$$
$$x \geq 3.97330384716953 \wedge x \leq 8.59306676718964$$
$$x \geq 10.2564891543491 \wedge x \leq 14.8762520743692$$
$$x \geq 16.5396744615287 \wedge x \leq 22.8228597687083$$
$$x \geq 27.4426226887284 \wedge x \leq 33.725807995908$$
$$x \geq 35.3892303830675 \wedge x \leq 40.0089933030876$$
$$x \geq 41.672415690247 \wedge x \leq 46.2921786102672$$
$$x \geq 47.9556009974266 \wedge x \leq 52.5753639174468$$
$$x \geq 54.2387863046062 \wedge x \leq 58.8585492246263$$
$$x \geq 60.5219716117858 \wedge x \leq 66.8051569189654$$
$$x \geq 71.4249198389855 \wedge x \leq 77.7081051461651$$
$$x \geq 79.3715275333246 \wedge x \leq 83.9912904533447$$
$$x \geq 85.6547128405042 \wedge x \leq 90.2744757605243$$
$$x \geq 91.9378981476837 \wedge x \leq 96.5576610677038$$
$$x \geq 98.2210834548633$$